Fractions have a long history in China, and the original forms of fractions are different from the present ones.
Later, India appeared a score representative similar to China's. Later, * * * people invented the fractional line, and the expression of the score became like this.
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction. The denominator is to divide an object into several parts, and the numerator is to take several parts.
Fractional symbolic scores are generated in the process of measurement and calculation respectively. In the process of measurement, it is a whole or a part of a unit; In the calculation process, when two numbers (integers) are divided and cannot be divided, a score is obtained.
In fact, scores have long been produced, and the knowledge about scores has also been recorded in the culture of ancient civilizations. The ancient Egyptians and Babylonians also marked scores, and the ancient Greeks used L ",for example, α L" =1,β L "= 2, γ L" = 3.
As for adding an apostrophe "'"to the upper right corner of the number, it means a fraction of that number. As for China, scores were adopted a long time ago. The earliest study of fractions in the world appeared in Nine Chapters of Arithmetic, in which fractions and their operations were systematically discussed.
("Arithmetic Nine Chapters Square Field Square Technique" points out: "The denominator is multiplied by others, and the numerator is followed. This formally gives the concepts of denominator and numerator).
There were two kinds of fractional notation in ancient China, one is Chinese notation, which is the same as the current Chinese notation: "… points …"; The other is the calculation notation: when calculating division, quotient comes first, real (that is, dividend) comes last, and method (that is, divisor) comes last. When the whole division is completed, the middle real number may remain, as shown in the figure, indicating the fraction. In the 3rd century AD, people in China used this notation to express scores.
For example, the notation of fractions in ancient India is very similar to that in China. /kloc-in the 20th century, * * * Hessel was the first to adopt the fraction.
Since then, he has been saying. Fibonacci was the first person to introduce scores into Europe.
It was not until the15th century that the modern fractional algorithm was gradually formed. 1530, Rudolf, a German, calculated+,and then gradually adopted the present fractional form.
1845, Augustus de Morgan put forward an oblique "/"in the article Calculus of Functions. Because the fraction is represented by a/b, which is beneficial to printing and typesetting, some printed books now use this oblique "/"fraction symbol.
I'll give you another website, and you can see it yourself /w? CT = 17 & amp； lm = 0 & ampbaiduWikiSearch & amppn = 0 & amprn= 10。 word=%B7%D6%CA%FD .
2. A little common sense of math achievement is urgently needed.
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.
The number representing this share is called the fractional unit. The maximum unit of a score is 1/2: Review the concept and definition of a score: divide the unit "1" into several equal parts, and the number representing such one or several parts is called a score, such as 1/2, 1/5, 2/6, 7/3. M stands for its number, that is, m, and the horizontal line in the middle of the molecule (diagonal line in this paper) is called this score, which is called "fractional line". The denominator n stipulates that it cannot be zero, …. When the above m is negative, m/n is negative, and the positive and negative scores are collectively called fractions.
The decimal unit is 1/n, when n= 1, 2, 3, 4, 5, 6, 7, 8, 9, ..., then 1/2, 1/3,1/. ..... are decimal units, that is, the numerator is 1 and the denominator is a fraction of a natural number equal to or greater than 2, which is called decimal units ... (When n= 1,/n =1=1. Obviously, 1/2 is a fraction, 1/2 is a decimal unit, and 1/2 is the largest decimal unit. Other fractions do not have this triple property-the decimal units of other ordinary fractions are less than 1/2. At the same time, a new concept of "decimal unit" is proposed.
Is the decimal unit, if you have the concept of decimal unit, you should have the largest decimal unit, because 1/2 is the largest decimal unit, because 1/2=0.5, then 0.5 is the largest decimal unit.
Obviously, 0.5 is a decimal, 0.5 is a decimal unit and 0.5 is the largest decimal unit. Other common decimals do not have this triple attribute-the decimal units of other common decimals are smaller than the maximum decimal unit, so 0.5 is the maximum decimal unit. (Editor: Qi Dong) 2 → numerator → fractional line 3→ denominator reading: two thirds Writing: 2-3 A horizontal line in the middle of the score is called the fractional line, the number above the fractional line is called the numerator, and the number below the fractional line is called the denominator.
Read it as a score. Fractions can be expressed by the division formula: for example, half equals 1 divided by 2.
Where 1 numerator is equal to dividend,-fractional line is equal to divisor, denominator of 2 is equal to divisor, and 0.5 fractional value is equal to quotient. Scores can also be expressed as ratios, for example; Half is equal to 1:2, where 1 numerator is equal to the previous paragraph, a fractional line is equal to the comparison number, the denominator of 2 is equal to the latter term, and the value of 0.5 points is equal to the ratio.
The basic nature of a fraction: the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, and the obtained fraction is equal to the original fraction. A/b=a/b=a:b(b is not equal to zero) Fractions also have an interesting property: Fractions are either finite decimals or infinite cyclic decimals, and infinite acyclic decimals like π cannot be replaced by fractions.
Another property of the fraction is that when the numerator and denominator are multiplied or divided by the same number at the same time, the fractional value will not change. Therefore, each score has an infinite number of equal parts.
Using this property, we can be on and off.
3. What is the most natural number?
The smallest natural number is 0.
Story:
Tang Priest and his disciples went to the Western Heaven to learn Buddhist scriptures. One day, they passed by Taoyuan and stopped to have a rest. The Monkey King and Zhu Bajie drool at the sight of peaches. The master said, "You can eat peaches, but I have to test you first." Wukong and Bajie nodded and said, "Yes, yes." The host said, "There are four peaches, and you two share them equally. How many peaches does everyone get? Please write down this number. " Hearing this, the apprentice laughed, which was not easy! Put pen to paper and wrote a "2". The master then said, "If you divide two peaches equally, how many will each person get?" ? Write down this number again. "the Monkey King load, conveniently wrote a" 1 ". The master said unhurriedly, "if you give a peach to both of you equally, how much money will each of you get?" "? How to write? ""half! " "Half!"
"How to write half?" Two disciples, you look at me, I look at you, at a loss.
Fractions have a long history in China, and the original forms of fractions are different from the present ones. Later, India appeared a score representative similar to China's. Later, * * * people invented the fractional line, and the expression of the score became like this.
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.
Denominator means to divide an object into several parts equally, and numerator means to take several parts.
4. Fractional knowledge
fractional symbols
fractional symbols
Scores are generated in the process of measurement and calculation respectively. In the process of measurement, it is a whole or a part of a unit; In the calculation process, when two numbers (integers) are divided and cannot be divided, a score is obtained.
In fact, scores have long been produced, and the knowledge about scores has also been recorded in the culture of ancient civilizations. The ancient Egyptians and Babylonians also marked scores, and the ancient Greeks used L ",for example, α L" =1,β L "= 2, γ L" = 3. As for adding an apostrophe "'"to the upper right corner of the number, it means a fraction of that number.
As for China, scores were adopted a long time ago. The earliest study of fractions in the world appeared in Nine Chapters of Arithmetic, in which fractions and their operations were systematically discussed. ("Arithmetic Nine Chapters Square Field Square Technique" points out: "The denominator is multiplied by others, and the numerator is followed. This formally gives the concepts of denominator and numerator). There were two kinds of fractional notation in ancient China, one is Chinese notation, which is the same as the current Chinese notation: "… points …"; The other is a preparation method:
When calculating division by calculation method, quotient comes first, real (dividend) is in the middle, and method (divisor) comes last. When the division is completed, the real number in the middle may have a remainder, as shown in the figure, indicating a fraction. In the 3rd century AD, people in China used this notation to express scores.
For example, the notation of fractions in ancient India is very similar to that in China. /kloc-in the 20th century, * * * hessel first adopted the fractional line. Since then, he has been saying. Fibonacci was the first person to introduce fractional lines into Europe. It was not until the15th century that the modern fractional algorithm was gradually formed. 1530, Rudolf, a German, calculated+,and then gradually adopted the present fractional form.
1845, in the article "calculus of functions", de Morgan proposed to use the diagonal line "/"to represent fractional lines. Because the fraction is represented by a/b, which is beneficial to printing and typesetting, some printed books now use this diagonal "/"fraction symbol.
5. The fifth grade math score application problem (50)
Application problems of mathematics scores in the fifth grade of primary school;
1 .20 black rabbits, and the number of white rabbits is 1/4. How many white rabbits are there?
2. There are 20 black rabbits, and the number of white rabbits is more than that of black rabbits 1/4. How many white rabbits are there?
3. There are 20 black rabbits, the number of which is 65438+ 0/4 of the number of white rabbits. How many white rabbits are there?
4. There are 20 black rabbits, and the number of black rabbits is more than that of white rabbits 1/4. How many white rabbits are there?
A duck weighs 3 kilograms, and the weight of a chicken is 2/3 of that of a duck. How much does this chicken weigh?
6. A volleyball is priced in 60 yuan, and the price of basketball is 5/6 of that of volleyball. What is the price of basketball?
7. There is 18 yuan in Xiao Liang's savings box, and Xiaohua's savings is 5/6 that of Xiao Liang. How much did Xiaohua save?
8. Xiaohong has 36 stamps, and the stamps in Xiao Xin are six fifths of Xiaohong's. How many stamps are there in Xiao Xin?
9. Grade six students collected 180 cans, which is 3/5 of that collected by grade five. How many cans did they collect in the fifth grade?
10, two children jump rope, Xiaoming jumped 100 times, Xiaoming jumped 5/8 of Xiao Qiang's jump, how many times did Xiaoming jump?
1 1, Xiaohong weighs 42kg, which is 2/3 of Xiao Ya's weight. What's the weight of Xiao Ya?
12, long-distance running exercise, Xiao Xiong ran 6 kilometers, which is 3/5 of that of Xiao Yong. How many kilometers did Xiao Yong run?
13, Xiao Wang read a book, reading 26 pages in the morning and reading 2/7 of the whole book. How many pages are there in the whole book?
6. Find the knowledge points about primary school math scores.
People's Education Press, Grade Five Mathematics Volume II
First, the significance and nature of music score
1 significance of the score
2 true score and false score
3. The basic nature of the score.
4 greatest common divisor and least common multiple
Five points. Turn fractions into common denominators
6 Reciprocity of Fractions and Decimals
Second, the addition and subtraction of scores
The significance of 1 decimal addition and subtraction.
Addition: the operation of combining two numbers into one number.
Subtraction: The operation of finding the other addend by knowing the sum of two addends and one of them.
2 calculation method and suddenness
(1) Addition and subtraction of fractions with the same denominator
Method: The denominator remains unchanged. Molecular addition and subtraction
(2) The scores of different denominators are added. negative
Methods: divide points first, then add and subtract.
(3) Fraction addition and subtraction mixed operation
(1) is calculated from left to right without brackets.
(2) If there are brackets, do it in brackets first, and then do it outside brackets.
(4) simple operation
Integer additive commutative law and associative law are also applicable to fractional addition.
?
7. Knowledge about fractions
More than 200 years ago, the Swiss mathematician Euler said in his book "General Arithmetic" that it is impossible to divide a 7-meter-long rope into three equal parts because there is no suitable number to represent it. If we divide it into three equal parts, each part is rice. Just like a new number, we call it a score.
Why is it called a score? The name score shows the characteristics of this number intuitively and vividly. For example, a watermelon is divided equally among four people. Why not divide it into four equal parts? As can be seen from this example, the score is generated by the need of measurement and mathematics itself-the need of division operation.
China is the first country to use scores. There are many records about scores in ancient China. According to Zuo Zhuan, during the Spring and Autumn Period, the largest vassal city could not exceed Zhou, and the middle city could not exceed Xiao.
During the Qin Shihuang period, the number of days in a year was 365.
Nine Chapters Arithmetic is a mathematical monograph written by China 1 0 more than 800 years ago. The first chapter, Square Domain, talks about four algorithms of fractions.
8. Knowledge points of fractions, percentages and decimals
(1) decimal
1, the meaning of decimal
Divide the integer 1 into 10, 100, 1000 ... a tenth, a percentage, a thousandth ... can be expressed in decimals.
One decimal place indicates a few tenths, two decimal places indicate a few percent, and three decimal places indicate a few thousandths. ...
Decimal system consists of integer part, decimal part and decimal part. The point in the number is called the decimal point, the number to the left of the decimal point is called the integer part, and the number to the right of the decimal point is called the decimal part.
In decimals, the series between every two adjacent counting units is 10. The propulsion rate between the highest decimal unit "one tenth" of the decimal part and the lowest unit "one" of the integer part is also 10.
2. Classification of decimals
Pure decimals: Decimals with zero integer parts are called pure decimals. For example, 0.25 and 0.368 are pure decimals.
With decimals: decimals whose integer part is not zero are called with decimals. For example, 3.25 and 5.26 are all decimals.
Finite decimals: The digits in the decimal part are finite decimals, which are called finite decimals. For example, 4 1.7, 25.3 and 0.23 are all finite decimals.
Infinite decimal: The digits in the decimal part are infinite decimal, which is called infinite decimal. For example: 4.33...3. 145438+05926 ...
Infinite acyclic decimal: the decimal part of a number with irregular arrangement and unlimited digits. Such decimals are called infinite cyclic decimals. For example: ∈
Cyclic decimal: the decimal part of a number, in which one or several numbers appear repeatedly in turn, is called cyclic decimal. For example: 3.555 … 0.0333 …12.15438+009 …
The decimal part of cyclic decimal is called the cyclic part of cyclic decimal. For example, the periodic part of 3.99 is "9" and the cyclic part of 0.5454 is "54".
Pure cyclic decimal: the cyclic segment starts from the first digit of the decimal part, which is called pure cyclic decimal. For example: 3.111.5656 ...
Mixed cycle decimal: the cycle section does not start from the first digit of the decimal part. This is called mixed cyclic decimal. 3. 1222……0.03333……
When writing a cyclic decimal, for simplicity, the cyclic part of the decimal only needs one cyclic segment, and a dot is added to the first and last digits of this cyclic segment. If there is only one number in the circle, just click a point on it. For example: 3.777 ... Jane writing 0.5302302 ... Jane writing.
(2) scores
1, the meaning of the score
Divide the unit "1" into several parts on average, and the number representing such a part or parts is called a fraction.
In the score, the middle horizontal line is called the dividing line; The number below the fractional line is called the denominator, indicating how many copies the unit "1" is divided into on average; The number below the fractional line is called the numerator, indicating how many copies there are.
Divide the unit "1" into several parts on average, and the number representing one part is called fractional unit.
2. Classification of scores
True fraction: The fraction with numerator less than denominator is called true fraction. The true score is less than 1.
False fraction: Fractions with numerator greater than denominator or numerator equal to denominator are called false fractions. False score is greater than or equal to 1.
With fraction: False fraction can be written as a number consisting of integer and true fraction, which is usually called with fraction.
3. Subtraction and total score
Changing a fraction into a fraction equal to it, but with smaller numerator and denominator, is called divisor.
The denominator of a molecule is a fraction of a prime number, which is called simplest fraction.
Dividing the scores of different denominators by the scores of the same denominator equals the original score, which is called the total score.
(3) Percentage
A number indicating that one number is a percentage of another number is called a percentage, also called a percentage or a percentage. Percentages are usually expressed as "%". The percent sign is a symbol indicating percentage.
Differences and connections:
Percent is another way to express a score. Percent is a fraction whose denominator is 100. Percentages cannot replace units, and fractions cannot replace units when expressing fractions, but they can replace units when expressing quantities.